Chapter 13: Applications of Hard Magnets
1.
Magnetic Circuits
2.
Materials
3.
Static Applications
4.
Dynamic Applications with Mechanical Recoil
5.
Dynamic Applications with Active Recoil
6.
Microsystems
Comments and corrections please: [email protected]
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Further reading
• R. Skomski and J. M. D. Coey, Permanent Magnetism, IOP, 1999
A monograph focussed on the physics of permanent magnetism, with chapters on experimental methods, materials and
applications.
• P. Campbell, Permanent Magnet Materials and their Applications, CUP, 1994 207 pp
A short and readable book for engineers.
• P. Abele, Stuctures of Permanent Magnets, Wiley, 1998
A monograph on magnet structures which generate static magnetic fields..
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Permanent magnets are ferromagnets with a wide hysteresis loop. Once magnetized, they sit at a
working point in the second quadrant of the loop which is determine by the magnet shape and the
rest of the magnetic circuit..
Slope of the load line is Bm/Hm
= -µ0(1/N - 1)
Permanent magnets generate magnetic flux with no continual expenditure of energy!
The B-field may be uniform or nonuniform, static or time-dependent. The magnetic flux density B0 in
the airgap is the natural field to consider in permanent magnet applications because flux is conserved
in a magnetic circuit, and forces on electric charges and magnetic moments all depend on B.
The best permanent magnets are intermetallic compounds of a ferromagnetic 3d element and a 4f
element; e.g. SmCo5 or Nd2Fe14B. Most common are the cheap hexagonal ferrites BaFe12O19 and
SrFe12O19. These powders are sometimes bonded in plastic
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Examples of permanent magnet applications.
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Field
Magnetic effect
Application
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Uniform
Nonuniform
Zeeman splitting
Torque
Hall effect,
Magnetoresistance
Force on conductor
Induced emf
motors, actuators, loudspeakers
generators, microphones
Forces on charged
particles
Force on paramagnet
Force on iron
Force on magnet
beam control, radiation sources
(microwave, uv, X-ray)
mineral separation
holding magnets
bearings, couplings, maglev
Time-varying Variable field
Force on iron
Eddy currents
magnetic resonance imaging
magnetic powder alignment
sensors
magnetometery
switchable clamps
brakes, metal separation
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Other uses of magnets, in acupuncture, pain control, electrochemistry, supression of wax formation in
oil wells or control of limescale deposits in pipes carrying hard water are difficult to classify, but worthy
of investigation !
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Applications depend on one of the following effects:
! A static uniform field
generates torque on a magnet and tends to align pre-existing magnetic
moments since " = m # B. The compass is an example.
Charged particles moving through a uniform field with velocity v are deflected by the Lorentz force F =
qv # B, which causes them to move in a helix. (c.f Busch’s e/m experiment)
! If the charged particles are electrons confined in a conductor of length L where they constitute a
current I flowing perpendicular to the field, and the Lorentz force leads to the familiar expression F =
BIL. This is the basis of electric motors and other drives.
Conversely, moving a conductor through the uniform field generates an induced electromotive force
(emf) given by Faraday's law! = -d$/dt where $ (=BA) is the flux threading the circuit of which the
conductor forms a part. Eddy currents are generated to oppose the motion.
! Spatially nonuniform fields offer another series of useful effects. They exert a force on a magnetic
moment given by the energy gradient F = - %(m.B).
They also exert nonuniform forces on moving charged particles, which can be used to focus ion or
electron beams or generate electromagnetic radiation from accelerating electron beams passing through
the nonuniform field.
!Time-varying fields can be produced by displacing or rotating the magnets. This may induce an emf in a
conductor according to Faraday's law exert forces on the induced eddy currents. Uniform time-varying
fields are valuable for magnetic measurements.
Finally, a spatially nonuniform time-varying field will exert a time-dependent force on a magnetic moment
or particle beam. Applications include magnetic switches and magnetic measurements such as the
Faraday balance.
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13.1 Magnetic Circuits
Assuming no flux leakage
BmAm = -BgAg
Assuming ideal soft material µ = &
Ampere’s law 'H.dl = 0
Hmlm = -Hglg
Multiplying BmHmVm = - Bg2Vg/µ0
Dividing -Bm/Hm = µ0 Aglm/ Amlg
The permeance coefficient
Figure 13.1 A simple magnetic circuit, and its electrical equivalent,
P = 1/R
with and without losses.
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13.1.1 Static and dynamic Applications
Figure 13.2 Hysteresis loops showing working points for a static application (a), a
dynamic application with mechanical recoil (b) and a dynamical application with
active recoil (c)
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13.2 Materials
H = -NM
B = µ0(H + M)
BH = -µ0(-NM + M)NM
= -µ0M2(N - N2)
d(BH)/dN = 1 - 2N = 0
N = 1/2
New icon for permanent magnets! (
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Motor
Loudspeaker
Figure 13.3 Influence of permanent magnet properties on the design of a dc motor
and a loudspeaker
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Tonnage production
Ferrite: 1,000,000
Nd-Fe-B 50,000
6 B$
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B
H
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BHmax < (1/4)µ0Ms2
11
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13.3 Static Applications
13.3.1 Uniform fields. The magnetic field produced by a point dipole of moment m Am2 is quite
inhomogeneous In polar coordinates, it is
Hr = 2m cos )/4!r3,
H) = m sin )/4!r3,
H* = 0
The field due to an extended line dipole of length L and dipole moment + Am per unit length is
significantly different:
Hr = + cos ),/4 ! r2,
H) = + sin )/4 ! r2,
Hz= 0
The magnitude of H, ,(Hr 2 + H)2 + H*2), is now independent of ) and its direction makes an angle 2)
with the orientation of the magnet.
Comparison of the magnetic field produced by a) a point dipole m and b) a line dipole +.
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Magnetic circuits made of long cylindrical segments may be used to generate uniform fields. An open
cylinder or a design with flat cuboid magnets and a soft iron return path is used to for nuclear magnetic
resonance (NMR). Permanent magnet flux sources supply fields of order 0.3 T with homogeneity of 1
part in 105 in a whole-body scanners.
Fig 13.5 Designs for magnetic cylinders which produce a uniform transverse field.
Figure (c) shows a design where the direction of magnetization of any segment at angular position - in
the cylinder is at 2- from the vertical axis. According to the equations for the line dipole, all segments
now contribute to create a uniform field across the airgap in a vertical direction. Unlike the structure of
Fig (a), the radii r1 and r2 can take any values without creating a stray field outside the cylinder. This
ingeneous device is known as a Halbach cylinder, The field in the airgap is
B0 = Br ln(r2/r1)
In practice it is convenient to assemble the device from n trapezoidal segments, as illustrated in fig. (d )
for n = 8.
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13.3.2 Nonuniform fields
Figure 13.6 Some cylindrical magnet structures which produce inhomogeneous
fields: (a) a quadrupole field (b) a hexapole field and (c) a uniform field gradient.
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Figure 13.7 A wiggler magnet used to generate intense electromagnetic radiation
from an electron beam.
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Nonuniform magnetic fields offer social benefits from the tiphead to the haematology laboratory. The
expression for the energy of a pre-existing magnetic moment m in a field H is - µom.H, leading to
F = µo%(m.H)
.
However when a small moment m = .V/ is induced by the field in a material of volume V and
susceptibility ., the expression becomes
F = (1/2)µo . V %(H2)
This expression is the basis of magnetic separation
Fig 13.8 Magnetic separation a) open gradient separation; b) electromagnetic separation with permanent magnets.
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Figure 13.9 Field and force patterns around a cylindrical iron wire in a high-gradient
magnetic separator.
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Figure 13.10 Permanent magnet variable flux sources: (a) a double Halbach cylinder.
(b) four–rod mangle.
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Figure 13.11 A MULTIMAG permanent-magnet variable flux source and controller.
The magnet head produces a variable field of up to 1.8 T in any transverse direction in
the bore.
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Figure 13.12 A vector vibrating-sample magnetometer based on a MULTIMAG
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Holding magnets
Magnets exert forces on each other, and on other ferromagnetic materials such as soft iron. The plasticbonded ferrite magnets on the ‘fridge are magnetized in a pattern of stripes about 3 mm wide
magnetized alternately inwards and outwards from the sheet.
Check this by gently dragging two pieces of
plastic magnet past each other !
! - ! -
! -
! -
Magnetization pattern of plastic magnet sheet.
To work out the maximum force that can be generated at the face of a magnet, consider a toroid that is
cut into two C-shaped segments and then separated slightly.
If the separation is d and the cross section area is Ag, the
energy appearing in the air gaps is 2#(1/2)µoHg2Agd
= Bg2Agd/µo. The work done separating the segment
is 2Fd, hence the force per unit area is
F/Ag = Bg2/2µo
Forces of up to 40 N cm-2 can be achieved for Bg = 1 T.
Fig 13.13 A magnetic toroid cut and separated to
produce a field in the airgap.
The flux density at the surface of the plastic magnet is about 50 mT. Estimate the force on a piece the
size of a credit card
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Figure 13.14 Two designs for switchable magnetic clamps. (a) is a rotatable magnet
design shown in the ‘on’ position, (b) is a design where the magnet array is displaced
laterally, shown in the ‘off’ position.
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Figure 13.15 (a) A face-type coupling with four axially-magnetized segments, (b) A
2:1 magnetic gear with radially-magnetised segments.
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Figure 11.16 Two elementary magnetic bearings made from axially-magnetised
rings; (a) a radial bearing and (b) an axial bearing.
Figure 11.17 A linear magnetic bearing.
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Figure 11.18 A Maglev system based on eddy-current repulsion.
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Figure 13.19 A magnetically-compensated hinge.
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Figure 13.20 Variable-reluctance sensor
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Figure 13.21 A flat voice-coil actuator for a personal-computer disk drive.
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Figure 13.21 Moving-iron actuators: (a) print hammer and (b) reed switch.
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Figure 13.22 DC motor designs: (a) brush motor with magnets on the stator and
(b) brushless motor with magnets on the rotor.
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Figure 13.23 Motors: (a) A two-pole dc brush motor, (b) a two-pole four-phase
brushless dc motor and (c) a variable reluctance motor.
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Figure 13.24 Variants pf the brushless dc motor: (a) normal design (b) cup-type (c)
disk-type. 1 – magnet; 2 – stator; 3 – stator winding; 4 – position sensor.
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Figure 13.24 A four pole synchronous motor with a permanent magnet rotor. A 16
bar squirrel cage winding is incorporated so that the machine will operate as an
induction motor for startup.
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Lavet Motor
Figure 13.25 A two pole stepping motor used in clocks and watches. In watches the
magnet made of bonded Sm2Co17 has a mass of a few mg.
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Figure 13.26 Miniature hybrid stepping motor
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Magnet applications; A 30 B" market
Others
Soft ferrite
Amorphous
Hard ferrite
Ni-Fe/Fe-Co
Hard
Magnets
Nd-Fe-B
Sm-Co
Fe-Si (oriented)
Alnico
Others
Soft
Magnets
Co- ! Fe 2 O 3
(tapes, floppy discs)
CrO2 (tapes)
Iron (tapes)
Fe-Si
Co-Cr (hard discs)
Magnetic
Recording
Iron
Others
Ni-Fe/Fe-Co (heads)
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